# Schwarzschild phase without a black hole

## TL;DR

This paper introduces a smooth, non-singular extension of the Schwarzschild solution that eliminates the black hole horizon and singularity, providing a new perspective on classical gravity without horizons.

## Contribution

It presents a novel vacuum solution in first-order gravity that avoids singularities and horizons, challenging traditional black hole models.

## Key findings

- No curvature singularity in the extended solution
- Absence of horizon and global time in the new geometry
- Negative mass Schwarzschild solution cannot be similarly extended

## Abstract

We present a smooth extension of the Schwarzschild exterior geometry, where the singular interior is superceded by a vacuum phase with vanishing metric determinant. Unlike the Kruskal-Szekeres continuation, this solution to the first-order field equations in vacuum has no singularity in the curvature two-form fields, no horizon and no global time. The underlying non-analytic structure provides a distinct geometric realization of `mass' in classical gravity. We also find that the negative mass Schwarzschild solution does not admit a similar extension within the first-order theory. This is consistent with the general expectation that degenerate metric solutions associated with the Hilbert-Palatini Lagrangian formulation should satisfy the energy conditions.